# A characterisation of F_q-conics of PG(2,q^3)

## TL;DR

This paper characterizes F_q-conics in PG(2,q^3) by relating them to 3-special normal rational curves in PG(6,q) through the Bruck-Bose representation, using geometric interactions between different projective space representations.

## Contribution

It provides a new geometric characterization of F_q-conics in PG(2,q^3) via their correspondence to 3-special normal rational curves in PG(6,q).

## Key findings

- F_q-conics correspond to 3-special normal rational curves in PG(6,q)
- Characterization of which normal rational curves relate to F_q-conics
- Use of geometric interactions between Bruck-Bose and Bose representations

## Abstract

This article considers an F_q-conic contained in an F_q-subplane of PG(2,q^3), and shows that it corresponds to a normal rational curve in the Bruck-Bose representation in PG(6,q). This article then characterises which normal rational curves of PG(6,q) correspond via the Bruck-Bose representation to F_q-conics of PG(2,q^3). The normal rational curves of interest are called 3-special, which relates to how the extension of the normal rational curve meets the transversal lines of the regular 2-spread of the Bruck-Bose representation. This article uses geometric arguments that exploit the interaction between the Bruck-Bose representation of PG(2,q^3) in PG(6,q), and the Bose representation of PG(2,q^3) in PG(8,q).